Stochastic Multi-Objective Short Range Fixed Head Hydrothermal Scheduling Using Classical, PSO & TLBO

Stochastic Multi-Objective Short Range Fixed Head Hydrothermal Scheduling Using Classical, PSO & TLBO

A.Surya Prakasha Rao, Associate Prof.

Department of Electrical & Electronics Engineering Sir C R Reddy College of Engineering

Eluru, India

M.vijay

Department of Electrical & Electronics Engineering Sir C R Reddy College of Engineering

Eluru, India

Abstract

This paper presents the multi-objective optimal solution for short range fixed head hydrothermal scheduling using classical method, Newton rapshon method and pso. The problem is formulated as a non-linear constrained multi-objective optimization problem. Considering the scheduling horizon period of 24 hours, hourly generation schedules are obtained for each of both hydro and thermal units for the three cases. The transmission losses are also accounted for through the use of loss coefficients. More no of simulations are carried out to obtain the best solution and the average value considered to improve the behaviour of pso. Numerical simulation of sample test system shows the effectiveness of the proposed algorithms. Index: pso, short range fixed head hydrothermal system, optimization technique, lagrangain relaxation.

1. Introduction

   In the present set-up of large systems with hydro and thermal power stations, the integrated operation of these power stations is inevitable and the economic aspect of such an operation cannot therefore be overlooked. The underlying idea of integrated operation is for optimum utilization of all energy sources in the most economical manner, so that an uninterrupted supply can be made available to the consumer. The operating cost of thermal plant is very high, though their capital cost is low. So it has become economical as well convenient to have both thermal and hydro plants in the same grid. The hydroelectric plant can be started quickly and it has higher reliability and greater speed of response. Hence hydroelectric

   plant can take up fluctuating loads. But the starting of thermal plants is slow and their speed of response is slow. Normally the thermal plant is preferred as a base load plant whereas the hydroelectric plant is run as a peak load plant. The short-term hydrothermal scheduling model has a time horizon of one week or one day with an hourly time interval..

2. HYDROTHERMAL SCHEDULING

   Optimal scheduling of power plant generation is the determination of the generation for every generating unit such that the total system generation cost is minimum while satisfying the system constraints. The objective of the hydrothermal scheduling problem is to determine the water releases from each reservoir of the hydro system at each stage such that the operation cost is minimized along the planning period. The operation cost includes fuel costs for the thermal units, import costs from neigh boring systems and penalties for load shedding.

   The HTC problem is usually solved by decomposition of the original problem into long, medium and short term problems each one considering the appropriate aspects for its time step and horizon of study.

   It is also essential to take into consideration two basic aspects of the hydro system:

   • The available water quantity (water inflows) is stochastic in nature.

   • The decision for the energy allocated to hydro units is deterministic.

3. Need of Hydrothermal Scheduling

   The operating cost of thermal plant is very high, though their capital cost is low. On the other hand the operating cost of hydroelectric plant is low, though their capital cost is high. So it has become economical as well convenient to have both thermal and hydro plants in the same grid. The hydroelectric plant can be started quickly and it has higher reliability and greater speed of response. Hence hydroelectric plant can take up fluctuating loads.

4. Short Range Problem

   The load demand on the power system exhibits cyclic variation over a day or a week and the scheduling interval is either a day or a week. As the scheduling interval of short range problem is small, the solution of the short-range problem can assume the head to be fairly constant. The amount of water to be utilized for the short-range scheduling problem is known from the solution of the long-range scheduling problem.

   A set of starting conditions (e.g. reservoir levels) is given, and the optimal hourly schedule that

   minimizes a desired objective, while meeting hydraulic steam, and electric system constraints, is sought.

   T

   qj t dt Vj

   0

   (j = 1, 2, , M)

   The short term hydrothermal scheduling problem is classified in to two groups

   • Fixed head hydro thermal scheduling

   • Variable head hydro thermal scheduling

     where Vj is the predefined volume of water available in cubic meters and hydro performance qj is represented by the conventional quadratic model, i.e.

     q t x P 2 t y P t z m3/h

     j 1 jN j jN j

     (j = 1, 2, , M)

5. HTS Problem formulation:

   Where xj, yj, and zj and the discharge coefficients of the jth hydro plant.

   5.4. Transmission Losses:

   N M N M N M

   N M N M N M

   A common approach to model transmission losses in the system is to use Krons approximated loss formula:

   PL t Pi t Bij Pj t Bi 0Pi t B00

   1. Objective function:

      The fixed-head hydrothermal problem can be defined considering the operating cost over the

      MW

      Where

      i1

      j1

      i1

      optimization interval to meet the load demand in each

      B00, Bi0, and Bij are B-coefficients.

      interval. Each hydro plant is constrained by the

      amount of water available for draw-down in the

      q jk

      is the rate of discharge from the jth hydro unit

      interval. The problem is defined as

      in interval k and is defined by

      Minimize

      q x P2 y P z m3/h

      T

      N

      T

      N

      J tk Fi Pik

      jk j jN ,k j j N ,k j

      N is the number of thermal units M is the number of hydro units

      Fi Pik

      k 1 i1

      is the cost function of thermal units in the

      T is the overall period for scheduling.

      The above objective as augmented by the

      interval k and is defined by

      2

      constraints is given as

      T N M

      T N M

      LP v t F P v t q

      P

      M N M

      • P P v V

      Fi Pik ai Pik bi Pik ci Rs/h

      ik , k , j

      k i ik j k jk k Dk Lk ik j j

      With ai, bi and ci as the cost coefficients.

      Pik is the output of thermal and hydro units

      k 1 i1 j1

      k k 1

      k k 1

      t Fi PLk 0

      i1 j1

      (i = 1, 2,..,N; k = 1, 2,

      during the kth interval.

      Pik

      Pik

   2. Constraints:

      ,T)

      q

      v t jk

      PLk

      0 (j = 1, 2, ,

      1. Load demand equality constraint:

         j k P k P 1

         N M

         mk mk

         M;

         Where

         Pi t PD t PL t

         i1

         m = N+ j; k = 1, 2, ,T)

         where

         is the incremental cost of power delivered in

         PD(t) is the load demand during the sub-interval.

         PL(t) is the transmission loss during the sub- interval.

      2. Limits are imposed as

         i i i

         i i i

         pmin P t pmax (i= 1, 2, , N + M)

   3. Volume and discharge:

   Eachhydro plant is constrained by the amount of water available for the optimization interval, i.e.

   the system during the kth interval.

   vj are the water conversion factor.

6. HTS methods:

1. Classical method:

   The problem we wish to set up is the general, short-term hydrothermal scheduling problem where the thermal system is represented by an equivalent unit, Psj. In this case, there is a single hydroelectric plant, PHj We assume that the hydro plant is not

   sufficient to supply all the load demands during the period and that there is a maximum total volume of water that may be discharged throughout the period of Tmax hours.

   6.1.1 Algorithm:

   i i i

   i i i

   1. Read the number of thermal units N, the number of hydro units M, the number of sub-intervals T, cost coefficients, a , b , c i 1, 2,…, N ,B-coefficients, Bij (i 1, 2,…, N M ; j 1, 2,…, N M ), discharge coefficients, xi , yi , zi i 1, 2,…, M ,demand PDk k 1, 2,…,T , the pre-specified available water Vj j 1, 2,…, M .

   2. Calculate the initial guess values of

      P0 i 1, 2,…, N M , 0 and v0 j 1, 2,…, M

2. PARTICLE SWARM OPTIMISATION:

Particle swarm optimization (PSO) is a population based stochastic optimization technique developed by Dr.Ebehart and Dr.Kennedy in 1995, inspired by social behaviour of bird flocking or fish schooling. PSO shares many similarities with evolutionary computation techniques such as Genetic Algorithms (GA). The system is initialized with a population of random solutions and searches for optima by updating generations. In PSO, the potential solutions, called particles, fly through the problem space by following the current optimum

particles. PSO has been successfully applied in many

ik k j

j

j

1. Consider v0 j 1, 2,…, M as calculated in Step 2.

2. Start the iteration counter, r = 1.

3. Start hourly count, k = 1.

6. Consider P0 i 1, 2,…, N M and 0 .

areas: function optimization, artificial neural network training, fuzzy system control, and other areas where GA can be applied. It mainly consists

three operations mutation, de acceleration and migration

1. Calculate

   ik

   Pik i 1, 2,…, N M

   k

   and k ,

   using the Newton-Raphson method. Gauss Elimination method is used to solve the following equations.

   Flow chart:

   k k P

   k

   pp p

   ik p

   k T

   0 k

   k

   p

2. Calculate the new values of Pik i 1, 2,…, N M

   and as Pnew P0 P and new 0

   k ik ik ik k k k

3. Set limits correspondingly as

   Pmax ;if Pnew Pmax

   i ik i

   ik i ik i

   ik i ik i

   Pnew Pmin

   ;if Pnew Pmin

   Pnew ;

   otherwise

   ik

   Disallow generator to participate, whose limits have been set either to lower or upper limit, in the scheduling by deleting that row and column.

   ik ik

   ik ik

4. Set P0 Pnew i 1, 2,.., N M and

   k k

   k k

   0 new GOTO Step 7 and repeat.

5. If

   k T , then GOTO Step 13, else k = k + tk,

   GOTO Step 6 and repeat.

   j

   j

6. Calculate water withdrawals V j 1,…., M .

   j j

   j j

7. If | V V s| or if (r R) then GOTO Step 14.

   else V new v0 V V s /V s ,

   j j j j j

   j j

   j j

   V 0 V new i 1, 2,…, M

   r r 1;

   GOTO Step 5 and repeat.

8. Calculate the optimal cost and loss and stop.

TLBO:

The TLBO algorithm is a newly Table-I. Analysis of cost & discharge values for these

developed meta-heuristic optimization algorithm [17]. It is a population-based optimization algorithm that is modelled based on the transfer of knowledge to the classroom environment, where learners first gain knowledge from a teacher (Teacher Phase) and then from fellow- students (Learner Phase). TLBO is a population based algorithm, where a group of students (i.e. learner) is considered the population and the different subjects offered to the learners are analogous with the different design variables of the optimization problem. The results of the learner are analogous to the fitness value of the optimization problem. The best solution in the entire population is considered as the teacher. The operation of the TLBO algorithm is explained below with the teacher phase and learner phase.

The structure of the proposed algorithm can be explicated as follows:

Step 1: Initializing the problem and algorithm parameters

Step 2: Establishing the initial population learners.

Step 3: Compute the objective function. Step 4: Compute the mean of the population.

Step 5: Determine the best solution (Teacher). Step 6: Modify solutions based on the teacher knowledge according to teacher phase.

Step 7: Update solutions according to learner phase and Steps 3.

Step 8: Go to Step 4 until the iteration number arrives at the maximum iteration number.

methods.

.

Table-I. Analysis of generations, losses and lamda values for these methods.

COMPARISON OF COST OF DIFFERENT METHODS:

The operating costs of classical, PSO and TLBO-method for 24 hours for given load demand. Cost obtained for TLBO less as compared with PSO and Classical methods. Comparison of cost for 24 hours

Table-III. Cost analysis for these methods.

Fig: cost analysis for these methods.

CONCLUSION AND FUTURE WORK:

The scheduling of electrical power from hydro and thermal plants has been done using PSO and classical methods. The optimization of cost is obtained mainly by employing three basic approaches. The optimization of thermal costs has been done by three methods employing programming technique. Finally the results of optimization by both the methods are tabulated and analysed. Numerical results show that highly near- optimal solutions can be obtained by T LB O. So it is clear that with the help of TLBO based algorithm it is possible to find a more nearly optimal solution to the existing hydrothermal scheduling problem.

Future works:

Hydro thermal scheduling problem with valve point loading can also be solved using EP. The valve point effect is modelled in two forms one is in the form of prohibited operating zones and the other is by including rectified sinusoidal component in the fuel cost function.

REFERENCES:

1). Power System Optimization by J.S.Dhillon & D.P.Kothari